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Theorem oppgmnd 14843
Description: The opposite of a monoid is a monoid. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.)
Hypothesis
Ref Expression
oppgbas.1  |-  O  =  (oppg
`  R )
Assertion
Ref Expression
oppgmnd  |-  ( R  e.  Mnd  ->  O  e.  Mnd )

Proof of Theorem oppgmnd
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oppgbas.1 . . . 4  |-  O  =  (oppg
`  R )
2 eqid 2296 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
31, 2oppgbas 14840 . . 3  |-  ( Base `  R )  =  (
Base `  O )
43a1i 10 . 2  |-  ( R  e.  Mnd  ->  ( Base `  R )  =  ( Base `  O
) )
5 eqidd 2297 . 2  |-  ( R  e.  Mnd  ->  ( +g  `  O )  =  ( +g  `  O
) )
6 eqid 2296 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
7 eqid 2296 . . . 4  |-  ( +g  `  O )  =  ( +g  `  O )
86, 1, 7oppgplus 14838 . . 3  |-  ( x ( +g  `  O
) y )  =  ( y ( +g  `  R ) x )
92, 6mndcl 14388 . . . 4  |-  ( ( R  e.  Mnd  /\  y  e.  ( Base `  R )  /\  x  e.  ( Base `  R
) )  ->  (
y ( +g  `  R
) x )  e.  ( Base `  R
) )
1093com23 1157 . . 3  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) )  ->  (
y ( +g  `  R
) x )  e.  ( Base `  R
) )
118, 10syl5eqel 2380 . 2  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) )  ->  (
x ( +g  `  O
) y )  e.  ( Base `  R
) )
12 simpl 443 . . . . 5  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  ->  R  e.  Mnd )
13 simpr3 963 . . . . 5  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  -> 
z  e.  ( Base `  R ) )
14 simpr2 962 . . . . 5  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  -> 
y  e.  ( Base `  R ) )
15 simpr1 961 . . . . 5  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  ->  x  e.  ( Base `  R ) )
162, 6mndass 14389 . . . . 5  |-  ( ( R  e.  Mnd  /\  ( z  e.  (
Base `  R )  /\  y  e.  ( Base `  R )  /\  x  e.  ( Base `  R ) ) )  ->  ( ( z ( +g  `  R
) y ) ( +g  `  R ) x )  =  ( z ( +g  `  R
) ( y ( +g  `  R ) x ) ) )
1712, 13, 14, 15, 16syl13anc 1184 . . . 4  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  -> 
( ( z ( +g  `  R ) y ) ( +g  `  R ) x )  =  ( z ( +g  `  R ) ( y ( +g  `  R ) x ) ) )
1817eqcomd 2301 . . 3  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  -> 
( z ( +g  `  R ) ( y ( +g  `  R
) x ) )  =  ( ( z ( +g  `  R
) y ) ( +g  `  R ) x ) )
198oveq1i 5884 . . . 4  |-  ( ( x ( +g  `  O
) y ) ( +g  `  O ) z )  =  ( ( y ( +g  `  R ) x ) ( +g  `  O
) z )
206, 1, 7oppgplus 14838 . . . 4  |-  ( ( y ( +g  `  R
) x ) ( +g  `  O ) z )  =  ( z ( +g  `  R
) ( y ( +g  `  R ) x ) )
2119, 20eqtri 2316 . . 3  |-  ( ( x ( +g  `  O
) y ) ( +g  `  O ) z )  =  ( z ( +g  `  R
) ( y ( +g  `  R ) x ) )
226, 1, 7oppgplus 14838 . . . . 5  |-  ( y ( +g  `  O
) z )  =  ( z ( +g  `  R ) y )
2322oveq2i 5885 . . . 4  |-  ( x ( +g  `  O
) ( y ( +g  `  O ) z ) )  =  ( x ( +g  `  O ) ( z ( +g  `  R
) y ) )
246, 1, 7oppgplus 14838 . . . 4  |-  ( x ( +g  `  O
) ( z ( +g  `  R ) y ) )  =  ( ( z ( +g  `  R ) y ) ( +g  `  R ) x )
2523, 24eqtri 2316 . . 3  |-  ( x ( +g  `  O
) ( y ( +g  `  O ) z ) )  =  ( ( z ( +g  `  R ) y ) ( +g  `  R ) x )
2618, 21, 253eqtr4g 2353 . 2  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  -> 
( ( x ( +g  `  O ) y ) ( +g  `  O ) z )  =  ( x ( +g  `  O ) ( y ( +g  `  O ) z ) ) )
27 eqid 2296 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
282, 27mndidcl 14407 . 2  |-  ( R  e.  Mnd  ->  ( 0g `  R )  e.  ( Base `  R
) )
296, 1, 7oppgplus 14838 . . 3  |-  ( ( 0g `  R ) ( +g  `  O
) x )  =  ( x ( +g  `  R ) ( 0g
`  R ) )
302, 6, 27mndrid 14410 . . 3  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
( x ( +g  `  R ) ( 0g
`  R ) )  =  x )
3129, 30syl5eq 2340 . 2  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
( ( 0g `  R ) ( +g  `  O ) x )  =  x )
326, 1, 7oppgplus 14838 . . 3  |-  ( x ( +g  `  O
) ( 0g `  R ) )  =  ( ( 0g `  R ) ( +g  `  R ) x )
332, 6, 27mndlid 14409 . . 3  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
( ( 0g `  R ) ( +g  `  R ) x )  =  x )
3432, 33syl5eq 2340 . 2  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
( x ( +g  `  O ) ( 0g
`  R ) )  =  x )
354, 5, 11, 26, 28, 31, 34ismndd 14412 1  |-  ( R  e.  Mnd  ->  O  e.  Mnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Mndcmnd 14377  oppgcoppg 14834
This theorem is referenced by:  oppgmndb  14844  oppggrp  14846  gsumwrev  14855  gsumzoppg  15232  gsumzinv  15233  oppgtmd  17796
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-0g 13420  df-mnd 14383  df-oppg 14835
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